Page 529 - Elements of Chemical Reaction Engineering Ebook

Sec. 8.6 Multiple Steady States 499

Bifurcation analysis will be applied to the CSTR mole and energy bal-
ances. First, Equations (8-68) and (8-69) are combined and the energy ballance
is written as
F(T) = Cpo (1 + K)T - Cpo (1 -k K)T, - G(T) (13-78)

which is of the form

F(T) = aT- p - G(T) (8-79)
Similarly, the CSTR mole balance can be written as

(13-80)

wl$ch is of a similar nature for the energy balance,

F(CA) = acA - p - G(cA) (8-81)
Both CSTR energy and mole balances are of the form

F(Y) = aY - P - G(Y) (8-82)
The conditions for uniqueness are then shown to be those that satisfy the
relatiolnship

max [ E)< a (8-83)

For example, if we use energy balance in the form given by Equation (8-78)
and use Equations (8-75) and (8-76) WE would find the criteria for uniqueness
(Le. No Multiple Steady States (MMS)) is

Criteria for no MSS

However, if

(E!-84)

we do not know if multiple steady solutions exist and we must carry the anal-
ysis further. Spexifically, the conditions for which multiple steady states exist
must satisfy the following set of equations:

(8-85)